Weak Galerkin methods for the Stokes eigenvalue problem

TL;DR

This paper develops a Weak Galerkin (WG) method for the Stokes eigenvalue problem, providing error estimates, lower bounds, and numerical validation within an abstract elliptic eigenvalue framework.

Contribution

It introduces a novel WG scheme with a new stabilizer and inequalities, offering theoretical error bounds and lower bounds for the Stokes eigenvalue problem.

Findings

The WG scheme achieves accurate eigenvalue approximations.

The method provides asymptotic lower bounds for eigenvalues.

Numerical examples confirm theoretical results.

Abstract

In this paper, we rewrite the Stokes eigenvalue problem as an Elliptic eigenvalue problem restricted to subspace, and introduce an abstract framework of solving abstract elliptic eigenvalue problem to give the WG scheme, error estimates and asymptotic lower bounds. Besides, we introduce a new stabilizer and several inequalities to prove GLB properties. Some numerical examples are provided to validate our theoretical analysis.